Studies of the acoustical properties of horns for loudspeakers have for a long time focussed on how a horn could enhance the sound pressure radiated by a loudspeaker, by acting as an acoustical transformer.
Direct radiating loudspeakers are known to be inherently inefficient due to the mismatch between the low acoustical impedance presented by the receiving medium (the air) and the relatively high mechanical impedance of the vibrating source (generally a moving diaphragm).
The fundamental theory of acoustical horns is based on Webster's equation, which describes the motion of an unidirectional wave inside a hollow body with rigid walls and slowly varying cross-section S(x):
                              ∂          2                ⁢        p                    ∂                  x          2                      +                  1        S            ⁢                        ∂          S                          ∂          x                    ⁢                        ∂          p                          ∂          x                      +                  k        2            ⁢      p        =  0
where p is the acoustic pressure, and k is the wave number
The end of the horn connected to the loudspeaker is referred to as the throat while the opposite end coupled to the ambient air is referred to as the mouth.
From Webster's equation, and assigning a particular mathematical function to the cross-section S along the propagation axis x, it is possible for a number of functions S(x) to derive the acoustical input impedance at the throat of the horn, if the radiating conditions at the mouth are known.
Analytical solutions for some specific functions are well known, for example in case of an exponentially varying cross-section:S(x)=S0·(e2.π·fc·x)2 
where S0 is the throat cross-section, and fc is the cut-off frequency of the horn.
The acoustic radiation impedance of an exponential horn, and few others like conical and hyperbolic horns can be found in reference works such as Olson (Acoustical Engineering, 1947), among others.
The exponential horn was long considered as an ideal choice because it exhibits a rapid though smooth rise in the acoustical throat impedance, thus achieving the expected gain in acoustic output from the lowest possible frequency.
On the other hand the conical horn was not rated so highly because of its poor loading characteristics at low frequencies.
However, there is another aspect to the properties of acoustical horns that had been overlooked in the early analysis which, as has been mentioned above, were mainly focused on efficiency and power output. In those days the electrical power delivered by amplifiers was limited to a few watts, a few tens at most.
Now, with modern power electronics, amplifiers can provide ample power for all applications, and the efficiency of the horn as an acoustic transformer is less of an issue, and more attention can be paid to horns as waveguides capable of controlling the directivity pattern of sound systems.
From this point of view exponential horns are certainly not ideal. This can be intuitively understood from the fact that the opening angle of an exponential horn varies greatly from the throat to the mouth: it is narrow at the throat and wide at the mouth. Relating this to the wavelength to radius ratio makes it easy to understand why the beamwidth of an exponential horn is wide at low frequencies and continuously narrows towards the high frequencies.
The conical horn having a constant opening angle from throat to mouth would seem to be the ideal candidate in terms of constant coverage. However, numerous experimental results have shown that this not the case. The typical behaviour of a conical horn shows a wide variation of beamwidth with frequency.
Often sound systems require different directivities in the horizontal and vertical planes. Hence a variation of the conical horn in the sectoral or radial horn: this has a constant but different opening angle, in the horizontal and vertical planes and hence a rectangular cross-section. However, radial horns inherit the shortcomings of conical horns (beaming), through not as acutely.
There have been numerous attempts to address the problem of better behaved and more constant directivity from acoustical horns, but none has been entirely satisfactory.